Answer: To determine the speed and direction of the white ball after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's denote the mass of the white ball as m1 and the mass of the black ball as m2, both of which are approximately 0.17 kg.
Given:
Mass of white ball (m1) = 0.17 kg
Mass of black ball (m2) = 0.17 kg
Initial velocity of white ball (u1) = 3.0 m/s
Final velocity of black ball (v2) = 1.5 m/s at an angle of 60° below the positive x-axis
First, let's calculate the momentum of each ball before and after the collision using the equation:
Momentum (p) = mass (m) x velocity (v)
Initial momentum of white ball (p1i) = m1 x u1
Final momentum of black ball (p2f) = m2 x v2
Using the given values, we have:
p1i = 0.17 kg x 3.0 m/s = 0.51 kg m/s
p2f = 0.17 kg x 1.5 m/s = 0.255 kg m/s (in the direction 60° below the positive x-axis)
According to the law of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. Therefore, we can write:
Total momentum before collision = Total momentum after collision
p1i = p1f + p2f
where p1f is the final momentum of the white ball after the collision.
Rearranging the equation to solve for p1f, we get:
p1f = p1i - p2f
Plugging in the values, we get:
p1f = 0.51 kg m/s - 0.255 kg m/s = 0.255 kg m/s
So, the final momentum of the white ball after the collision is 0.255 kg m/s.
Next, let's calculate the kinetic energy of each ball before and after the collision using the equation:
Kinetic energy (KE) = 0.5 x mass (m) x velocity^2 (v^2)
Initial kinetic energy of white ball (KE1i) = 0.5 x m1 x u1^2
Final kinetic energy of black ball (KE2f) = 0.5 x m2 x v2^2
Using the given values, we have:
KE1i = 0.5 x 0.17 kg x (3.0 m/s)^2 = 0.765 J
KE2f = 0.5 x 0.17 kg x (1.5 m/s)^2 = 0.19125 J
According to the law of conservation of kinetic energy, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision. Therefore, we can write:
Total kinetic energy before collision = Total kinetic energy after collision
KE1i = KE1f + KE2f
where KE1f is the final kinetic energy of the white ball after the collision.
Rearranging the equation to solve for KE1f, we get:
KE1f = KE1i - KE2f
Plugging in the values, we get:
KE1f = 0.765 J - 0.19125 J = 0.57375 J
So, the final kinetic energy of the white ball after the collision is 0.57375 J.
Now, let's determine if the collision is elastic or inelastic. In an elastic collision, both momentum and kinetic energy are conserved, meaning that the total momentum and total kinetic energy of the system before the collision should be equal to the total momentum and total kinetic energy of the system after the collision.
In this case, we can see that both momentum and kinetic energy are conserved, as calculated earlier. The total momentum before the collision is 0.51 kg m/s (in the direction of the initial velocity of the white ball) and the total momentum after the collision is 0.255 kg m/s (in the direction 60° below the positive x-axis for the black ball, and in the opposite direction for the white ball). Also, the total kinetic energy before the collision is 0.765 J and the total kinetic energy after the collision is 0.57375 J.
Since both momentum and kinetic energy are conserved, we can conclude that the collision is elastic.
Finally, to determine the speed and direction of the white ball after the collision, we can use the calculated final momentum of the white ball, which is 0.255 kg m/s in the opposite direction of its initial velocity. The magnitude of the velocity can be found by dividing the final momentum by the mass of the white ball, m1:
Final velocity of white ball (v1f) = p1f / m1
Plugging in the values, we get:
v1f = 0.255 kg m/s / 0.17 kg = 1.5 m/s
So, the speed of the white ball after the collision is 1.5 m/s, and it moves in the direction opposite to its initial velocity.